Discussion

may generate β energy-dependent contributions to the correlation coefficients that mimic the effects of the MSSM-induced scalar and tensor interactions discussed here. An analy- sis of these effects on the correlation coefficientsa andAhas been recently performed in Ref. [131]. To our knowledge, no such study has been carried out for the correlation coeffi- cients of interest here. Carrying out such an analysis, as well as sharpening the theoretical expectations of Eq. (9.45), would clearly be important for the theoretical interpretation of future correlation studies.

The effects of triscalar couplings in weak decay correlations arise from one-loop graphs that generate scalar and tensor interactions. These interactions are forbidden in the SM CC interaction in the limit of massless fermions since it involves only LH fermions and since the scalar and tensor operators couple fields of opposite chirality. In the MSSM, such terms can arise via L-R mixing of virtual scalar fermions in one-loop box graphs, and this L-R mixing can be significant when the corresponding soft, triscalar couplings are unsuppressed. In the case of µ-decay, additional contributions to scalar and tensor four-fermion operators can also be generated by flavor-mixing among same-chirality scalar leptons, but this flavor-mixing is highly constrained by LFV studies such asµ→eγ. Thus, for bothµ- and β-decay, observable, SUSY-induced scalar and tensor couplings can only be generated by flavor diagonal L-R mixing.

Probing these interactions would require improvements in precision of one- and two- orders of magnitude, respectively, forβ-decay andµ-decay correlation coefficients. Order of magnitude improvements forβ-decay appear realistic, while the necessary advances for µ-decay appear to be more daunting. On the other hand, if SUSY is discovered at the LHC, then considerations of SUSY-induced, four-fermion scalar interactions involving RH charged leptons may become necessary when extracting the Fermi constant from the muon lifetime. Doing so could become particularly important when ppm tests of electroweak symmetry become feasible.

Chapter 10 Conclusions

Electroweak-scale supersymmetry, if realized in nature, has many important implications for nuclear physics, particle physics, and cosmology. Supersymmetric electroweak baryo- genesis may explain the origin of the baryon asymmetry. We studied how the charge transport dynamics of collisions and diffusion play an important role in determining the BAU. Gaugino, strong sphaleron, and third generation Yukawa interactions are the most important interactions that convert hypercharge, generated within the bubble wall, into left- handed quark and lepton charge that drives baryon number generation. We evaluated the gaugino and third generation Yukawa thermally-averaged interaction rates for decay and absorption processes in the plasma. We found:

• Gaugino interactions are generally in chemical equilibrium for gaugino massesm_Ve . 1TeV. These interactions enforce superequilibrium — chemical equilibrium between a particle and its superpartner.

• Top Yukawa interactions are always in chemical equilibrium due to the large scatter- ing rateq₃H_u ↔u₃g. In addition, the decay processHe ↔q¯₃t_Rcan be even larger, further enhancing chemical equilibrium.

• Bottom and tau Yukawa interactions can be in chemical equilibrium for large regions of parameter space. For example, the decay processes Hd ↔ q3d¯3(`3¯e3) are in chemical equilibrium fortanβ &5(15), andm_A.800GeV (600 GeV).

These interaction rates enter into the system of Boltzmann equations that governs the charge densities.

Next, we solved the Boltzmann equations for the charge densities. Our main result was to show how the resulting charge densities depend strongly on the inclusion of bottom and tau Yukawa interactions, which had been previously neglected. We found:

• Bottom Yukawa interactions suppress the conversion of charge from third generation quarks to the first and second generations, via strong sphalerons. Without bottom Yukawa interactions, strong sphalerons generate significant first and second genera- tion left-handed quark charge.

• Bottom Yukawa interactions lead to a suppression of third generation left-handed quark charge when (i)metR, mebR & 500 GeV, or (ii) metR ' mebR. Without bottom Yukawa interactions, this suppression does not occur.

• Tau Yukawa interactions allow for the generation of significant left-handed lepton charge; without them, no lepton charge is generated.

One interesting possibility that emerges is that the baryon asymmetry is “lepton-mediated,”

i.e., induced by left-handed lepton charge, rather than left-handed quarks, as previously considered. Phenomenologically, the baryon asymmetry in this scenario can differ in both magnitude and sign from what one would have computed neglecting these Yukawa inter- actions. To the extent that electric dipole moment searches and collider studies can give information about the CP-violating phases and supersymmetric spectrum, these interac- tions play a crucial in making connections with experiment.

Lastly, we investigated how supersymmetry can be studied experimentally through pre- cision measurements of weak decays. Leptonic pion decay is sensitive to R-parity viola- tion in the first and second generations, the mass splitting between left-handed electron and muon scalar superpartners, and the Higgsino-Wino spectrum. Deviations from the SM expectation in precision studies of muon and beta decays can arise for large tri-scalar, left-right mixing parameters. If a supersymmetric signal was observed through these tests, it would point to regions of supersymmetric parameter space beyond the minimal SUSY- breaking scenarios to which theoretical prejudice has been mostly confined.

Appendix A

From Green’s Functions to Boltzmann Equations: an Overview of the

Closed-Time-Path Formalism

The standard tool for discussing particle dynamics in the early universe is the Boltzmann equation. The purpose of this section is to show how to derive it using the Closed-Time- Path (CTP) formulation of quantum field theory, and apply these techniques to supersym- metric electroweak baryogenesis. The CTP formalism is a language of finite-temperature, non-equilibrium Green’s functions. These propagators are powerful tools: they contain all the information about the dynamics of the theory — from the microscopic interactions to the macroscopic evolution of the thermal plasma. After a brief review of this formalism, we describe how to extract this information, and how our results related to standard treatments of the Boltzmann equation (see, e.g., Ref. [36]).

To be concrete, we will consider the example of a single complex scalar field ϕ, with Lagrangian

L=|∂_µϕ|²−m²_ϕ|ϕ|²+L_int . (A.1) We definefϕ(k,X, t)andfϕ¯(k,X, t)to be the particle and antiparticle distribution func- tions, for 3-momentumk, positionX, and timet ≡ X⁰. The distribution function for the charge density isf ≡fϕ−fϕ¯; its Boltzmann equation is

∂f

∂t + k

ω_k · ∇_Xf =C[f_ϕ, f_ϕ¯], (A.2)

with collision termC, a functional offϕandfϕ¯. (We have neglected Hubble expansion and external forces.)

Ultimately, it turns out that there are two classes of interactions that arise from the col- lision term in Eq. (A.2). First, there are the usual elastic and inelastic collisions of particles in the plasma. The interactions in Eqs. (2.26,2.27) are all of this type. The collision terms from these interactions are the usual thermally-averaged interaction rates; here, there is little benefit for using the CTP approach. Second, there are interactions that arise from the presence of the expanding bubble of broken electroweak symmetry, giving rise to a CP-violating source. Here, the CTP approach is essential.

A.1 Closed-time-path Green’s functions

At zero temperature, perturbation theory is essentially the study of time-ordered propaga- tors, such as

G^t(x, y) = D

T n

ϕ_H(x)ϕ^†_H(y) o E

, (A.3)

whereϕ_H is the Heisenberg-picture field. The key difference when moving to finite tem- perature is that the expectation value in Eq. (A.3) is taken with respect, not to the vacuum, but to the thermal bath, an ensemble of states defined by a density matrix

ˆ ρ≡X

n

w_n|n_hihn_h|, (A.4)

where the time-independent Heisenberg-picture states|n_hihave weightw_n.

Now, let us move to the interaction picture. First, the interaction-picture states |n(t)i are functions of time; we define the interaction states at timet=−∞to coincide with the Heisenberg states: |n−i ≡ |n(−∞)i =|nhi. The density matrix isρˆ= P

nwn|n−ihn−|.

Second, the interaction fields ϕare related to their Heisenberg counterparts by the time- evolution operatorU

ϕ_h(x) =U(x₀,−∞)^†ϕ(x)U(x⁰,−∞). (A.5)

The operatorU obeys the usual relations:

U(t₁, t₂) =U(t₂, t₁)^† =U(t₂, t₁)⁻¹ (A.6) and

U(t₁, t₂) =T

½ exp

µ i

Z _t2

t1

dz⁰ Z

d³zL_int(z)

¶ ¾

. (A.7)

With these relations, Eq. (A.3) becomes

G^t(x, y) = X

n

w_n D

n₋

¯¯

¯T

½ exp

· i

Z

d⁴zL_int(z)

¸¾_†

× T

½

ϕ(x)ϕ^†(y) exp

· i

Z

d⁴zL_int(z)

¸¾ ¯¯

¯n₋ E

, (A.8) whereR

d⁴z =R_∞

−∞dz⁰R

d³z. Reading from right to left, Eq. (A.8) corresponds to starting with the “in”-state|n₋i, then time-evolving from−∞to +∞, acting with the field oper- ators at times x⁰ andy⁰ along the way, and lastly time-evolving from +∞back to −∞, returning to the “in”-state. This time-contour, denoted C, is the “closed time path”; it is closed in the sense that the contour begins and ends at t = −∞, connecting “in”-states with “in”-states. Eq. (A.8) can then be succintly written as

G^t(x, y) =

¿ P

½

ϕ+(x)ϕ^†₊(y) exp

· i

Z

C

d⁴zLint(z)

¸¾À

=

¿ P

½

ϕ+(x)ϕ^†₊(y) exp

· i

Z d⁴z

³

L⁽⁺⁾_int(z)− L⁽⁻⁾_int (z)

´¸¾À

(A.9) whereP meanspath-ordering of fields alongC. In the second line, we have brokenC into the sum of its two branches. The notationϕ±(x)andL^(±)_int(x)— itself a function ofϕ±(x)

— denotes whether x⁰ is on the time-increasing (+), or time-decreasing (–) branch of C. The path-ordering prescription is to time-order the (+) fields, to anti-time-order (T^†) the (–) fields, and lastly to put all the (–) fields to the left of the (+) fields.

A perturbative evaluation ofG^t(x, y)proceeds similarly to zero-temperature field the- ory. Wick’s theorem applies as usual, but withP-ordering instead of T-ordering. There- fore, we must consider not one but four different propagators, corresponding to all possible

path-ordering ofx⁰ andy⁰inhϕ(x)ϕ^†(y)i:

G^>(x, y) ≡ D

P n

ϕ₋(x)ϕ^†₊(y) o E

=­

ϕ(x)ϕ^†(y)®

(A.10a) G^<(x, y) ≡

D P

n

ϕ₊(x)ϕ^†₋(y) o E

=­

ϕ^†(y)ϕ(x)®

(A.10b) G^t(x, y) ≡

D P

n

ϕ₊(x)ϕ^†₊(y) o E

=­ T©

ϕ(x)ϕ^†(y)ª ®

(A.10c)

= θ(x⁰−y⁰)G^>(x, y) +θ(y⁰−x⁰)G^<(x, y) G^¯t(x, y) ≡

D P

n

ϕ₋(x)ϕ^†₋(y) o E

=­ T^†©

ϕ(x)ϕ^†(y)ª ®

(A.10d)

= θ(y⁰ −x⁰)G^>(x, y) +θ(x⁰−y⁰)G^<(x, y).

These Green’s functions are the free or full propagators for fields in the interaction- or Heisenberg-pictures, respectively. In particular, we see from Eq. (A.9) that a perturbative expansion of G^t(x, y)will necessarily involve contracting (+) and (–) fields together (i.e., those contained inL⁽⁻⁾_int); these additional propagators are inescapable. These propagators can be assembled into the matrix

G(x, y)e ≡



 G^t(x, y) −G^<(x, y) G^>(x, y) −G^¯t(x, y)



 . (A.11)

In this form, the propagators have the simple perturbative expansion G(x, y) =e Ge⁽⁰⁾(x, y) +

Z d⁴w

Z d⁴z

³Ge⁽⁰⁾(x, z)eΠ(z, w)G(w, y)e

´

(A.12a) G(x, y) =e Ge⁽⁰⁾(x, y) +

Z d⁴w

Z d⁴z

³G(x, z)e eΠ(z, w)Ge⁽⁰⁾(w, y)

´

. (A.12b) These equations are the CTP version of the Schwinger-Dyson equations, where now both the propagator and the self-energy eΠ(defined byL_int) are2×2matrices. The free propa- gatorGe⁽⁰⁾(x, y)satisfies

¡∂_x²+m²_ϕ¢ Ge⁽⁰⁾(x, y) = ¡

∂_y²+m²_ϕ¢ Ge⁽⁰⁾(x, y) = −i δ⁴(x−y)I ,e (A.13) whereIedenotes the2×2identity matrix in CTP propagator space.

In vacuum, the Green’s functions G^<,>(x, y) depend only on the relative coordinate r ≡ x−y, through plane wave factors e^±ik·r, with frequency ω_k and wavelength |k|⁻¹. At finite temperature, however, expectation values are taken with respect to the thermal plasma. The plasma itself is dynamical — namely, distribution functions for species in the plasma depend on spacetime — so G^<,>(x, y) depends also the average coordinate X = (x+y)/2. The Wigner transform, defined as

G^<,>(k, X) = Z

d⁴r e^ik·rG^<,>(x, y), (A.14) naturally separates the microscopic dynamics (relevant over scalesω_k⁻¹, |k|⁻¹ ∼ O(T⁻¹)) from the macroscopic evolution of the plasma. This separation between macro- and mi- croscopic scales is valid only as long as the plasma dynamics is characterized by scales much longer thanT⁻¹. In our analysis, we rely upon this separation of scales to perform a gradient expansion; formally denoted as an expansion in∂_X, it is essentially an expansion in the ratio of micro- to macroscopic scales.

The Wigner-transformed propagators in Eq. (A.14) are the functions of primary interest.

In particular, we define the function F(k, X)≡ 1

2 (G^>(k, X) +G^<(k, X) ) . (A.15) The CP-asymmetric distribution functionf =f_ϕ−f_ϕ¯from Eq. (A.2) is

f(k, X) = Z dk⁰

2π 2k⁰F(k, X). (A.16) In addition, we can define “moments” of this distribution function. The zeroth moment is the charge density

n(X)≡

Z d³k

(2π)³ f(k, X) =

Z d⁴k

(2π)⁴ 2k⁰F(k, X), (A.17) the difference between particle and antiparticle densities. The first moment is the charge

current

j(X)≡

Z d³k (2π)³

k ωk

f(k, X) =

Z d⁴k

(2π)⁴ 2kF(k, X). (A.18) Justification of these relations follows from the fact that

Z d⁴k

(2π)⁴ 2k^µF(k, X) = i ­

:ϕ^†(X)^↔∂^µ_Xϕ(X) :®

, (A.19)

using equations (A.10, A.15). The normal-ordered combination of fields on the RHS is precisely the currentj^µ(X)≡(n,j)^µ. Lastly, the second moment is a velocity flux tensor

V_ab(X)≡

Z d³k (2π)³

k_ak_b

ω_k² f(k, X) =

Z d⁴k

(2π)⁴ 2k_ak_b

k⁰ F(k, X), (A.20) with spatial indeces a, b = 1,2,3. In what follows, our goal is to derive and solve a Boltzmann equation forF(k, X).